This section is intended to introduce various aspects of the art, which may be associated with embodiments of the disclosed techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the disclosed techniques. Accordingly, it should be understood that this section is to be read in this light, and not necessarily as admissions of prior art.
Reservoir models with a highly detailed geological and/or petrophysical description often include a number of gridblocks on the order of 1-10 million. Running a flow simulation on such large models is usually computationally prohibitive. Accordingly, an original geological model with a detailed description, denoted as a ‘fine scale model,’ should be coarsened to a ‘coarse scale model’ with much fewer numbers of gridblocks so that the reservoir model can be flow-simulated within a feasible computational time. The coarsening of the reservoir model involves regenerating a grid system, which is denoted as upgridding, and regenerating petrophysical properties (such as porosity and/or permeability) assigned to gridblocks, denoted as upscaling. This upgridding and upscaling should reasonably preserve original understandings of hydrocarbon in-place, estimates of reserves, and predicted production performance of the original fine scale model through the coarsening so as not to mislead business decisions. However, while the preservation of volumetric quantities such as hydrocarbon in-place is relatively straightforward, the preservation of dynamic quantities, such as recovery factors and/or production performance, is not a trivial task. For example, if the mechanism of hydrocarbon recovery is mainly due to displacement of hydrocarbons by other fluids, which are either injected from the surface or are encroaching from an aquifer, simple static calculation of coarse scale petrophysical properties (e.g. weight-averaging of fine scale properties within the coarse gridblock) is not a sufficient upscaling approach to preserve recovery factors and/or production performance. This is because hydrocarbon production due to the injection of displacing fluids (or due to water encroachment from an aquifer) is highly dependent on the efficiency of sweep, which is often controlled not only by large scale geological features but also by the small scale heterogeneity of petrophysical properties. If an important small scale feature of petrophysical heterogeneity is erased or smoothed by upscaling, the efficiency of sweep is miscalculated by the coarse scale flow simulation. To avoid this, upscaling of relative permeability, in addition to the upscaling of absolute permeability, is needed. Relative permeability is a measure of degree of ease of flow of individual phases of fluid, such as hydrocarbon liquid phase (e.g., oil) and aqueous phase (e.g., water), when two or more fluids are flowing simultaneously though porous media. Relative permeability is expressed as a function of phase saturation and is defined as a ratio of permeability of individual phases under multiphase flow condition to absolute permeability under single-phase flow condition. Thus the role of relative permeability in reservoir simulation is to control the amount of fluids of individual phases that flow across a gridblock interface in accordance with the phase saturation at an upstream gridblock.
The most commonly used method for upscaling relative permeability is known as the dynamic pseudo-function method. This method approximates the fine scale simulation of multiphase flow behavior by running a flow simulation on a coarsened model. The method generates and uses upscaled relative permeability as a pseudo-function that accounts for the effect of sub-grid scale heterogeneity on sweep. In other words, the method is a way to upscale rock relative permeability, which is measured by core flooding experiments on a small sample of rock and usually only representing characteristics of multiphase flow through a homogeneous porous medium, to upscale relative permeability that represents characteristics of multiphase flow through heterogeneous medium at the scale of a simulation gridblock. This method can approximate three-dimensional flow simulation using a two-dimensional areal model. However, it is still used as a method for upscaling relative permeability to coarsen a fine scale geologic model to a coarse scale flow simulation model. Numerous variations of dynamic pseudo-function methods have been proposed. One way to classify these variations is based on the boundary conditions used for generating the pseudo-function. Different boundary conditions generate different pseudo-functions for the same model. Each of the existing variations has its own limitations.
Cross-Sectional Simulation Based Method.
This boundary condition was used to reduce a three dimensional simulation model to a two dimensional areal model (i.e., a reservoir model that comprises only 1 layer). Pseudo-functions are generated from a fine scale simulation on a two or three dimensional cross-sectional model that represents a typical vertical cross-section of the reservoir. The limitation of this approach is that it is usually difficult to identify a ‘representative’ cross-section, if such a cross-section exists at all. This approach has been replaced by global methods as dynamic pseudo generation using full field scale fine simulation (“global methods”) became relatively affordable with three dimensional grids. However, because of its simplicity, it is still used occasionally to test new ideas and/or methods.
Global Method.
This boundary condition is supported by commercial software, such as the PSEUDO software package offered by Schlumberger. In this method, dynamic pseudo-functions are generated by post-processing the result of fine scale simulation run at the full field scale, using given well positions and rates. Therefore, the boundary condition used for generating pseudo-functions is the same as the condition that occurs at the simulation gridblocks in the given fine scale simulation. The limitation of this approach is that if well positions and rates are changed, the pseudo-functions should be regenerated by re-running the fine scale simulation. This can be a challenge when running the fine scale simulation is not computationally feasible.
Local Method.
Local methods generate relative permeability pseudo-functions by running two-phase flow simulation on a local domain as depicted in FIG. 1. This method can be seen as a dynamic pseudo-function generation method implemented in such a way that the flow simulation on the local domain mimics core flooding experiments. As shown in FIG. 1, the local domain 100 comprises an upstream coarse grid domain 102 adjacent a downstream coarse grid domain 104. Direction of fluid flow is shown by arrows 105a, 105b. A coarse grid interface 106 defines the interface between the upstream and downstream coarse grid domains. Modeled fluid flow through local domain is shown as shaded squares 107. A pseudo-function is constructed through post-processing of the simulation result by relating displacing phase saturation within the upstream coarse grid domain 102 to the relative permeability calculated from the flux of individual phases flowing across the coarse grid interface 106 and the pressure difference between upstream and downstream coarse grid domains 102, 104.
A typically used boundary condition is a constant pressure boundary condition which imposes constant pressure to the inlet and outlet faces 108, 110 of local domain 100. A displacing phase saturation of Sd=1.0 is imposed on inlet face 108 and Sd=0.0 is imposed on outlet face 110. A no-flow boundary condition is applied on the sides 112, 114 of local domain 100. The limitation of this approach is the discrepancy between the saturation boundary condition imposed on the local domain and the actual phase saturation that occurs in the related simulation model. The boundary condition of Sd=1.0 occurs in actual flow simulations only if the injector is placed in an adjacent simulation gridblock and never happens otherwise.
To overcome this limitation, many boundary condition variations are proposed. The Effective Flux Boundary Condition (EFBC) applies a constant flux boundary condition, instead of constant pressure, on inlet and outlet faces in such a way that the flow rate is allocated to the fine grids in accordance with fine scale permeability on the faces. Although EFBC improves the accuracy of reproduction of fine scale simulation compared to the constant pressure boundary condition, it still fails to capture the flow properties for cases with highly stratified heterogeneity structure. A possible reason is that EFBC still uses saturation boundary condition of Sd=1.0 at the inlet face, which is unrealistic.
Local-Global Method.
Local-global methods are proposed to avoid the need of running fine scale simulation at the full field scale. The sequential upscaling method sequentially generates pseudo-functions for simulation gridblocks, from upstream to downstream, using a local method by specifying flux rates of individual phases as an inlet boundary condition, instead of using constant pressure and a saturation boundary condition. The inlet flux rates are carried over from the two-phase simulation result that is previously conducted on the upstream local domain. In other words, the flux boundary condition on the local domain is specified in such a way that outflux of upstream coarse gridblock is the same as influx of downstream coarse gridblock. The limitation of this approach is that pseudo-functions are generated sequentially for all simulation gridblocks individually using a local method, which is computationally prohibitive in real reservoir cases.
A local-global two-phase upscaling method first runs a coarse scale two-phase simulation at the full field scale using rock relative permeability, and then utilizes the simulated coarse scale inter-block flux and saturation as boundary conditions for generating pseudo-functions for individual simulation gridblocks using a local method. If needed, the whole process is iterated by updating relative permeability by the previously generated pseudo-functions. As with the sequential upscaling method, the limitation of this approach is significant computational cost because, to iterate the process, the method requires pseudo generation for all individual simulation gridblocks by running the two-phase flow simulation on individual local domains.
The local-global methods described above rely on global methods and simply attempt to avoid running fine scale simulations at the full field scale. Therefore, the limitations of global methods also apply to local-global methods, i.e. if well positions and rates are changed, the pseudo-functions must be regenerated by re-running the fine scale simulation.
Extended Local Method.
Pickup et al. have proposed to use an extended local boundary condition to generate a pseudo-function as a form of tensor phase permeability. In this approach, which is shown in FIG. 2, the local domain 200 for pseudo-function generation is defined by including both 1) a coarse gridblock domain 202 where the pseudo-function is to be calculated and 2) the coarse gridblocks 204, 206, 208, 210, 212, 214, 216, 218 surrounding coarse gridblock domain 202. In this way, the boundary conditions are not imposed directly on the faces of the coarse gridblock where the pseudo-function is to be calculated. The limitation of this approach is that, because the pseudo-function is generated as a form of tensor—a phase permeability tensor—it should be used by flow simulation using a nine-point flux scheme, which is much more computationally expensive than conventional flow simulation. To use this tensor pseudo-function with conventional flow simulators using a two-point flux scheme, off-diagonal elements of the tensor should be omitted. However, the pseudo-function generated by this method loses the accuracy of reproduction of fine scale simulation if the off-diagonal elements of the tensor are neglected.
What is needed is a local method that solves the problems related to saturation boundary conditions of Sd=1.0, imposed on an inlet face of a gridblock, without the need of a tuning parameter or a significant increase in computational cost.